If the name change is to take the spouse's last name, it usually isn't necessary to follow any complex legal process; just file I-90 with the fee and a copy of the marriage certificate to change the name on the green card. The new card will take a number of months to be delivered, so it would be advisable to get the passport stamped with I-551 so it can be used to travel or work while the new card is in process ... but then that means the passport should be changed before filing I-90, so it will have the new name when going for stamping.
Then once the new green card is delivered, use that to change the Social Security card, bank account, driver's license, etc.
However, looking at your signature I see the interview has not happened yet. In that case, there is an even simpler route. At the interview, just show the marriage certificate and ask for the GC to be produced with the married name.
If the name change is other than taking the spouse's last name, it would be necessary to follow a legal process through the courts in your jurisdiction. Contact your local City Hall to find out what the forms and fees and process steps are. In some areas it is cheap and simple; in other areas it is complex and costs hundreds of dollars and requires appearing in court.